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PRACTICAL  TREATISE 


RAILWAY    CURVES 


AND 


LOCATION, 

FOR  YOUNG  ENGINEERS 


MtTAINING  A  FTTLL   DESCRIPTION  OF  THE  INSTRUMENTS,  THE  MANNER  OF  ADJUSTING   THEM,  AND 
THE    METHODS    OF    PROCEEDING    IN   THE   FIELD, — NEW  AND   SIMPLE   FORMULAE   FOR  COM- 
POUND   AND    REVERSE   CURVING,— RULES    FOR    CALCULATING   EXCAVATION   AND 
EMBANKMENT, — STAKING    OUT  WORK,  4C.,  TOGETHER  WITH   TABLES  OF 
NATURAL  SINES  AND  TANGENTS,  RADII,  CHORDS,  ORDINATES, 
AND  OTHERS  OF  GENERAL  USE  IN  THE  PROFESSION. 


WILLIAM   F.^SHUNK, 


crvn,  ENCUCEER. 


PHILADELPHIA : 
ENRY    CAREY     BAIRD, 

INDUSTRIAL    PUBLISHER. 

No.    406    WALNUT    STREET. 

18CG. 


Entered,  according  to  the  Act  of  Congress,  in  the  year  1854,  by 
E.  H.  BUTLER  A  00., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  in  and  for  the  Eastern 
District  of  Pennsylvania, 


r 

<* 


PREFACE. 


THE  located  line  for  railway  is  a  series  of  curves  and 
straight  lines,  or  tangents.     These  are  first  plotted  to  a 
large  scale  from  data  gathered  on  preliminary  survey.    It 
is  therefore  desirable  that  all  explorations  should  be  made 
with  extreme  care,  as  upon  their  correctness  depend,  in  no 
small  degree,  the  labour  and  time  required  in  location. 
It  were  better  for  accuracy  that  all  angles  should  be  made 
and  recorded  from  the  plates,  and  the  needle  used  only  as 
a  test,  or  check.     Good  chaining  is  indispensable.     Great 
attention,  too,  should  be  given  to  the  proper  use  of  the 
;  slope  instrument.     By  these  means  a  working  map  can  be 
'  constructed  in  the  oflice  upon  which  the  proposed  location, 
^  grade  lines,  &c.,  may  be  traced  with  tolerable  resemblance 
HEb  fact.     Still  many  errors  attach  to  both  data  and  map, 
•  and  these,  together  with  the  unexpected  obstacles  encoun- 
tered in  the  field,  require  ready  knowledge  of  the  means 
for  overcoming  them. 

It  has  been  my  design  to  present  this  knowledge  to  my 
younger  fellows  in  the  profession.     I  have  endeavoured  to 
*  %  i-~*-  ,««..>.  *-+.  r~v  \  ) 


IV  PREFACE. 

do  it  lucidly  and  concisely — without  supposing  unusual  cases 
— without  prolix  proof  or  complex  figuring.  The  problems 
given  are  of  frequent  occurrence,  and  the  tables  appended 
will  be  found  useful  and  correct. 

To  STRICKLAND  KNEASS,  a  gentleman  whose  professional 
abilities  are  well  known,  I  return  thanks  for  valuable 
assistance.  I  would  likewise  make  my  acknowledgments, 
for  useful  suggestions,  to  CHARLES  DELISLE,  an  engineer 
of  high  mathematical  attainment. 

I  am  aware  that  much  more  might  have  been  said — 
much  more  suggested — on  the  subject  of  location ;  but  a 
field  book  being  the  object,  the  compact  plan  precluded 
any  extensive  essay. 

If  the  work  with  brevity  combines  clearness,  and  ia 
comprehensive  withal,  it  is  the  work  intended. 

W.  F.  SHUNK. 


CONTENTS. 


PACK 

PREFACE       ........  3 

Explanations      ........  7 

ARTICLE  I. — Of  the  Instruments.     The  Level        .  .  .  ,11 

Its  adjustment     .  .  .  .  .  .  13 

The  Rod          .......      14 

Levelling  .  .  .  .  .  .  15 

The  Transit    .  .  .  .  .  .  .17 

The  Vernier          ......  18 

Adjustment  of  Transit  .  .  .  .  .19 

II. — Preliminary  propositions  ....  20 

III. — To  avoid  an  obstacle  in  tangent        .  .  .  .21 

IV.— Triangulation        ......  22 

V. — Of  calculating  tangents  to  any  degree  of  curvature  .      24 

VI. — To  trace  a  curve  with  transit  and  chain  ...  25 

VII. — To  triangulate  on  a  curve      ....  29 

VIII. — To  change  the  origin  of  any  curve,  so  that  it  shall  termi- 
nate in  a  tangent,  parallel  to  a  given  tangent       .  .      31 
IX. — To  change  a  P.  C.  C.  with  similar  object.    First.  When  the 

second  curve  has  the  smaller  radius  .  .  .32 

X. — Second.  "When  the  last  curve  has  the  larger  radius        .  34 

Synopsis  of  formula;  for  compound  curving          .  .      35 

XI. — Having  located  a  compound  curve  terminating  in  any  tan- 
gent, to  find  the  P.  C.  C.  at  which  to  commence  another 
curve  of  given  radius  which  shall  terminate  in  the  same 
tangent.    First.  When  the  latter  curves  have  the  smaller 
radii       .......  3f 

XII. — Second.  When  the  latter  curves  have  the  larger  radii          .      38 


VI  CONTENTS. 

PACI> 

ART.  XIII. — To  change  a  P.  R.  C.  so  that  the  second  curve  shall  termi- 
nate in  a  tangent  parallel  to  a  given  tangent        .  .      39 
XIV. — How  to  proceed  when  the  P.  C.  is  inaccessible               •  41 
XV. — To  avoid  obstacles  in  the  line  of  curve         .            .            .43 
XVI. — To  calculate  reverse  curves          ....  45 

XVII. — Having  given  a  located  curve,  terminating  in  any  given 
tangent,  to  find  where  a  curve  of  different  radius  will  ter- 
minate in  a  parallel  tangent  .  .  .  .40 

XVIII. — Having  a  curve  located  and  terminating  in  a  given  tangent, 
to  find  the  P.  C.  C.  whereat  to  begin  another  curve  of 
given  radius  which  shall  terminate  in  a  parallel  tangent       48 
XIX.— To  locate  a  Y  .  .  .  .  .  49 

XX. — To  run  a  tangent  to  two  curves         .  .  .  .51 

XXI.— Of  ordinates          ......  52 

XXII. — To  find  the  radius  corresponding  to  any  chord  and  deflexion 

angle.     Deflexion  and  tangential  distances    .  .  54 

XXIII. — Of  excavation  and  embankment        .  .  .  .56 

XXIV.— Side  staking         ......  60 

TABLES. 

Natural  Sines  and  Tangents      ......  63 

Radii .  .  .     89 

Long  Chords      .  .  .  .  .  ..90 

Ordinates      .........      91 

Squares  and  Square  Boots        ...•,.  94 

Slopes  and  Distances  for  Topography         •  •  •  >  .    106 


EXPLANATIONS. 


ALL  railway  curves  are  parts  of  circles.  They  are 
designated  generally  from  their  character  as  simple,  com- 
pound, or  reverse  ;  and  specifically  from  the  central  angle 
subtended  by  a  chord  of  100  feet  at  the  circumference, 
this  being  the  length  of  the  chain  in  common  use.  It  is 
found  that  the  circle  described  with  radius  of  5730  feet 
has  a  circumference  of  36,000  feet.  Since  there  are  360° 
in  the  circle,  the  central  angle  subtended  by  a  chord  of 
100  feet  is,  in  this  case,  equal  to  1°,  and  the  curve  is 
named  a  one  degree  curve.  So  likewise  in  a  circle  with 
radius  of  2865  feet,  half  of  5730,  the  central  angle  cor- 
responding to  the  chord  100  is  2°;  the  curve  is  then  called 
a  two  degree  curve. 

The  beginning  of  a  curve  is  called  the  point  of  curva- 
ture, or  simply  the  P.  C.,  and  its  termination  the  point  of 
tangent,  marked  P.  T. 

A  compound  curve  is  composed  of  two  curves  of  different 
radii,  turning  in  the  same  direction,  and  having  a  common 
tangent  at  their  point  of  meeting.  This  point  is  called  the 
point  of  compound  curvature,  or  P.  C.  C. 

(7) 


8  RAILWAY   CURVES   AND   LOCATION. 

A  reverse  curve  is  composed  of  two  curves  turning  in 
different  directions,  and  having  a  common  tangent  at  their 
point  of  meeting,  which  latter  is  named  the  point  of  re- 
versed curvature,  or  the  P.  R.  C. 

All  sines  and  tangents  made  use  of  in  this  work  are 
from  the  table  at  the  end  of  the  volume.  For  calculating 
curves  it  is  not  necessary  to  use  more  than  four  decimals. 

A  Bench  is  a  shoulder  hewn  with  the  axe  on  the  but- 
tressed base  of  a  tree,  and  so  shaped  at  the  top  as  to  afford 
footing  to  the  rod.  The  tree  is  blazed  and  the  elevation 
of  the  bench  marked  on  it  with  red  chalk.  Benches  serve 
as  permanent  reference  points  to  the  level.  They  are 
placed,  where  it  is  possible,  about  one  thousand  feet 
apart. 

Points.  The  operation  termed  pointing  is  the  fact  of 
putting  a  peg  firmly  into  the  ground,  and  of  driving  in  its 
top  a  tack,  or  making  thereon  an  indentation  whose  place 
is  indicated  by  cross  keel  marks,  directly  in  the  line  of  col- 
limation  of  the  transit.  Thus  true  lines  are  traced  on  the 
ground,  and  angles  measured  accurately.  When  the  transit 
is  set  over  a  point  it  is  so  posited  that  the  plumb  hangs 
immediately  above  the  tack  head.  If  the  head  plate  of 
the  tripod  be  much  inclined  the  plumb  should  be  examined 
after  levelling  the  instrument,  as  that  operation  disturbs  it 
to  some  extent. 

Stations.  The  line  of  a  survey  is  marked  on  the  ground 
at  regular  intervals,  by  stakes  two  feet  in  length,  blazed, 
and  numbered  from  0  up  in  arithmetical  progression. 
These  stakes  are  named  stations.  On  exploration  they 
are  commonly  placed  two  hundred,  and  on  location,  one 
hundred  feet  apart. 

It  is  customary,  when  locating,  to  drive  pegs  even  with 
the  surface  along  the  true  line,  and  to  place  the  stations  a 
couple  of  feet  to  the  right,  numbers  facing  in,  to  show  their 


EXPLANATIONS.  9 

position.  The  pegs  are  less  liable  to  disturbance  from  frost, 
animals,  &c. 

In  locating  for  construction  stakes  are  driven  on  sharp 
curves  at  intervals  of  50,  sometimes  25  feet. 

The  Chain  in  general  use  for  railway  surveys  is  made 
of  soft  iron.  It  is  100  feet  long,  and  divided  into  100 
links,  each  one  foot  in  length.  At  every  tenth  link  ia 
attached  a  brass  drop,  toothed  so  as  to  indicate  its  distance 
from  the  end.  It  presents  the  advantages  of  durability, 
accuracy,  and  expedition. 


A  PRACTICAL  TREATISE 


RAILWAY  CURVES  AND   LOCATION. 


ARTICLE  I. 

OF    THE    INSTRUMENTS. 
THE   LEVEL. 

level  is  an  instrument  used  in   ascertaining  the 
undulations  of  the  ground 
along  the  line  of  a  survey,    rv,  — "-      «  l 

and  of  measuring  these  ir-  ^ 
regularities  accurately  in 
reference  to  an  assumed 
base  called  the  datum.    It 
consists    mainly    of    the 
telescope  k  i,  the  spirit- 
level  and  its  encasement 
5,  the  Y's  c  c,  the  rectan- 
gular bar  o  d,  the  axis  e,  the  plates  and  levelling  screws 
/  ?»,  and  the  tripod  g, 

In  the  tube  of  the  telescope  at  7i,  and  at  right  angles  to 
its  axis,  is  placed  a  flat  ring,  called  the  diaphragm.  To 
this  ring  the  cross-hairs  are  attached — two  delicate  spider 
lines  stretched  over  it  vertically  and  horizontally,  and  inter- 
secting at  the  centre.  It  is  held  in  position  by  four 

(11) 


12  RAILWAY   CURVES   AND   LOCATION. 

slightly  movable  screws,  which  pierce  the  tube  in  the 
direction  of  the  "  cross-hairs."  i  is  a  milled  head  for 
adjusting  the  focus  of  the  object  glass,  and  k  an  inserted 
tube,  containing  several  lenses,  which  may  be  moved  out 
or  in  so  as  to  make  the  spider-lines  distinctly  visible. 

A  straight  line  looked  along  from  the  eye  glass  at  k 
through  the  intersection  of  the  cross-hairs  is  the  line  of 
sight,  technically  named  the  line  of  collimation, 

The  immediate  supports  of  the  telescope  are  called  the 
Y's,  from  their  resemblance  to  that  letter.  If  a  small  arch 
were  sprung  between  the  two  legs  of  the  Y  it  would  give 
a  good  idea  of  the  clasping  pieces  which  hold  the  telescope 
in  place.  They  are  jointed  to  One  leg  and  secured  to  the 
other  by  pins  which  may  be  withdrawn  and  the  pieces 
turned  back  in  order  to  remove  the  telescope,  or  change  it 
end  for  end. 

The  Y's  are  attached  at  right  angles  to  the  bar  d,  which, 
again,  is  connected  firmly  at  right  angles  with  the  hollow 
axis  e.  This  latter  fits  closely  over  and  is  revolvable  hori- 
zontally around  a  solid  axis  »,  which,  passing  through  the 
plate  /,  is  secured  to  the  head  of  the  tripod  by  means  of  a 
loose  ball-and-socket  joint.  The  plate /has  four  levelling 
screws  inserted  in  it ;  with  these  the  instrument  may  be 
brought  to  a  horizontal  position  even  when  the  lower  plate 
is  considerably  inclined. 

One  of  the  Y's  is  movable  for  a  short  space  up  or  down 
by  means  of  the  capstan-head  screw  o.  The  spirit-level  is 
likewise  movable  both  vertically  and  laterally  by  means  of 
screws  at  either  end. 

n  is  a  clamp  screw,  and  p  a  tangent-screw  for  slightly 
turning  the  telescope  in  a  horizontal  direction. 


THE   LEVEL,  13 

To  adjust  the  Lewi* 

First.  To  make  the  line  of  collimation  coincide  with  the 
xxis  of  the  telescope. 

Set  the  instrument  firmly,  and  direct  the  telescope 
toward  some  distant,  distinct  object,  such  as  a  nail-head. 
Clamp  fast,  and  with  tangent-screw  fix  the  line  of  collima- 
tion upon  the  object  accurately.  Revolve  the  telescope 
half  way  round  in  the  Y's,  *'.  e.  until  the  bubble  is  above 
it,  and  if  the  horizontal  spider-line  still  covers  the  point,  it 
requires  no  adjustment.  If  it  does  not,  reduce  the  error 
one-half  by  means  of  the  diaphragm  screws,  and  complete 
the  reduction  with  the  capstan-head  screw.  Revolve  the 
telescope  round  to  its  first  position,  and  if  the  horizontal 
line  and  point  do  not  then  coincide,  repeat  the  operation 
until  they  do,  in  any  position  of  the  telescope.  In  similar 
wise  the  vertical  hair  may  be  adjusted,  when  the  line  of 
collimation  should  cover  the  point  through  an  entire  revo- 
lution of  the  telescope. 

Great  care  should  be  taken  in  this  as  well  as  in  all  other 
adjustments  of  cross-hairs,  that  the  opposite  screw  of  the 
diaphragm  be  loosened  before  tightening  its  fellow,  or 
injury  to  the  instrument  must  result. 

Second.  To  make  the  axis  of  the  spirit-level  parallel  to 
the  line  of  collimation. 

With  levelling  screws  bring  the  bubble  to  the  middle  of 
its  tube,  reverse  the  telescope  in  its  Y's,  and  if  the  bubble 
does  not  then  stand  in  the  middle  correct  one-half  the 
deviation  with  the  screw  at  the  left  end  of  the  bubble-case, 
and  the  other  half  with  the  capstan-head  screw.  Again 
reverse  the  telescope  in  its  Y's,  and,  if  necessary,  repeat 
the  operation. 

Now  revolve  the  telescope  a  short  distance  in  its  Y's,  so 
as  to  bring  the  spirit-level  to  one  side  of  its  lowest  position. 
If  the  bubble  deviates  from  the  middle,  correct  the  error 

2 


14  RAILWAY    CURVES    AND    LOCATION. 

with  the  lateral  screws  at  the  right  end  of  the  bubble-case, 
and  examine  the  previous  adjustment  before  lifting  the 
instrument. 

Third.  To  bring  the  line  of  collimation  parallel  to  the 
bar. 

Turn  the  telescope  until  it  stands  directly  over  two  of 
the  levelling  screws,  and  with  them  bring  the  bubble  to  the 
middle  of  the  tube.  Then  revolve  the  telescope  horizontally 
until  it  stands  over  the  same  screws,  changed  end  for  end. 
If  the  bubble  does  not  still  stand  in  the  middle  of  the  tube, 
correct  one-half  the  deviation  with  the  capstan-head,  and 
one-half  with  the  levelling  screws. 

Place  the  telescope  over  the  other  levelling  screws  and 
proceed  in  a  similar  manner,  and  continue  the  corrections 
until  the  bubble  stands  without  varying  during  an  entire 
revolution  of  the  instrument  upon  its  axis. 

This  completes  the  adjustment  of  the  level. 

THE   ROD. 

The  rod  used  in  levelling  consists  of  a  staff  and  a  target, 
which  latter  is  so  attached  to  the  staff  as  to  be  movable 
along  it  from  end  to  end.  The  rod  is  commonly  seven  feet 
long,  but,  being  composed  of  two  rectangular  pieces  fitted 
together  by  means  of  a  sliding  groove,  it  can  be  extended 
to  nearly  double  that  length.  It  is  graduated  to  feet  and 
tenths  of  a  foot.  The  target  is  a  circle  of  wood  or  iron, 
usually  four-tenths  in  diameter,  and  divided  into  quadrantal 
sectors  by  a  horizontal  and  vertical  line  which  intersect  at 
its  centre.  The  sectors  are  painted  alternately  red  and 
white,  so  that  their  dividing  lines  are  visible  at  a  consider- 
able distance.  On  the  back  of  the  target,  where  it  meets 
the  graduated  side  of  the  rod,  is  fixed  a  chamfered  brass 
edging,  whereon  the  space  of  one-tenth  is  graven  from  the 
centre  down.  This  is  subdivided  into  ten  spaces  marking 


LEVELLING. 


15 


hundredths,  and  these  latter  divided  into  halves,  so  that  the 
height  of  the  middle  of  the  target  above  the  base  of  the 
rod  may  be  accurately  read  to  within  -005  of  a  foot. 

There  is  a  similar  graduated  tenth  on  the  standing  part 
of  the  rod,  to  be  used  for  high  sights  when  the  sliding 
groove  comes  into  play. 

Both  target  and  rod  are  provided  with  clamp  screws. 

LEVELLING. 

The  operation  technically  called  levelling  is  performed 
thus : — 

Suppose  a  the  starting  point,  or  zero,  in  reference  to 
which  all  the  inequalities  of  the  surface  along  the  line  of 
survey  are  measured,  as  at  the  points  c,  e,  f.  The  hori- 
zontal line  a  /is  called  the  datum  line.  This  is  arbitrarily 


assumed.  It  may  be  considered,  for  example,  at  any  dis- 
tance above  the  point  a,  and  the  irregularities  of  the  ground 
measured  from  an  imaginary  level  line  in  ether ;  but  for 
convenience  of  figuring,  and  other  politic  reasons,  it  is  cus- 
tomary in  seaport  towns  to  take  high  tide  as  datura.  In- 
land, the  summer  surface  of  the  nearest  stream,  or,  when 
commencing  on  a  ridge,  the  highest  neighbouring  knoll  is 
assumed. 

Well !  suppose  a  to  be  zero,  and  the  instrument,  for 
instance,  set  and  levelled  at  b.  Stand  the  rod  at  a,  and 
slide  the  target  up  until  its  cross-lines  are  covered  by  the 
cross-hairs  in  the  telescope  ;  i.  e.,  until  the  line  of  collima- 
tion  coincides  with  the  centre  of  the  target.  The  leveller 
directs  the  movements  of  the  target  by  raising  or  lowering 


16  RAILWAY   CURVES  AND   LOCATION. 

his  hand.  A  circular  motion  of  the  hand  signifies  "  make 
fast."  The  bubble  should  always  be  examined  before  the 
rod  is  taken  down,  and  the  latter  should  be  read  twice,  or, 
if  convenient,  shown  to  the  leveller,  in  order  to  guard 
against  mistake.  If  in  this  case  it  reads  8  feet,  the  height 
of  the  instrument  is  then  8  feet  above  a.  To  find  the  ele- 
vation of  o  above  «,  take  the  rod  thither  and  lower  the 
target  until  coincidence  results  as  before.  If  the  rod 
reads  2  feet,  of  course  e  is  8  —  2  =  6  feet  above  a. 

If  it  is  necessary  to  lift  the  instrument  here,  a  small  peg 
is  driven  at  c  before  sighting  to  that  point,  to  insure  firm 
footing  for  the  rod.  Sighting  back  from  the  new  position, 
d,  the  rod  reads  5  feet ;  then  5  +  6,  the  elevation  of  c 
above  a,  =  11,  the  height  of  the  telescope  at  d  above  a. 
If  at  e  the  reading  is  6,  the  elevation  of  that  point  is  11 
—  6  =  5,  and  if  at  /  the  reading  is  8  the  elevation  of 
that  point  in  reference  to  a  is  11  —  8  =  3,  marked  -f-  3. 

The  rule,  therefore,  in  levelling  is,  at  each  new  stand  of 
the  instrument,  to  add  the  reading  of  the  rod  sighted  back 
at,  to  the  discovered  elevation  of  the  point  at  which  the 
rod  stands,  for  the  height  of  the  instrument ;  and  to  sub- 
tract from  this  height  the  reading  of  the  rod  at  any  points 
observed  from  the  new  position  in  order  to  find  the  eleva- 
tion of  those  points.  The  above  is  noted  in  the  field-book 
as  follows : 


Station. 

Rod. 

Height  of 
Instrument. 

Total,  or 
Elevation. 

a 

8-00 

8-00 

00 

c 

2-00 

+  6.00 

5-00 

11-00 

e 

6-00 

-f  5-00 

f 

8-00 

-f  3-00 

OP    THE    TRANSIT.  17 

The  advantages  of  this  method  of  levelling  over  tkf  old 
system  of  backsights  and  foresights  are,  that  it  affords 
readier  facilities  for  testing  the  correctness  of  the  work, 
and  it  may  be  carried  on  more  rapidly.  By  the  old  plan 
each  sight  at  the  rod  was  linked  with  that  which  preceded 
it,  and  added  one  more  to  a  continuous  calculation  in  which 
a  single  error  affected  all  the  following  work.  Here,  how- 
ever, if  haste  is  required,  the  calculation  of  the  interme- 
diate sights  or  "cuttings"  may  be  omitted  entirely  while 
in  the  field,  the  reading  of  the  rod  only  being  set  down ; 
the  "totals"  may  be  worked  from  peg  to  peg,  and  the  lia- 
bilities to  mistake  thus  decreased  about  eighty  per  cent. 

OF   THE   TRANSIT. 

The-  transit  is  an  instrument  for  measuring  horizontal 
angles.  It  consists  of  the  tele- 
scope a  c,  the  Y's  (i,  the  compass-  .A  ^C  ^P$~" ^ 
box,  &c.,  e  g,  and  the  axia  k. 
The  telescope  is  furnished  like 
that  of  the  level,  and  the  instru- 
ment is  similarly  fitted  to  its  tri- 
pod. The  telescope  revolves  in 
a  vertical  circle,  and  is  attached 
to  the  Y's  by  means  of  a  trans- 
verse axis  whose  extremities  turn  in  smooth  journals  at  the 
head  of  the  Y's.  The  body  of  the  instrument  at/  contains 
a  magnetic  needle,  with  its  usual  circular  surrounding, 
graduated  to  degrees  and  quarter  degrees.  The  flooring  of 
this  box  has,  on  one  side,  an  opening  with  chamfered  edge 
upon  which  the  vernier  is  engraved.  This  latter,  together 
with  the  telescope,  Y's,  and  all  the  upper  part  of  the 
instrument,  is-made  to  revolve  by  means  of  the  screw  /*, 
upon  a  solid  plate  beneath,  which  is  likewise  graduated 
from  0  to  180°  each  way.  Thus  angles  may  be  measured 

2* 


18  RAILWAY   CURVES   AND   LOCATION. 

accurately  without  using  the  needle  at  all.  It  need  be 
regarded  merely  as  a  check,  g  is  a  clamp  screw  for  secur- 
ing the  plates  together,  and  i  a  screw  for  fastening  the 
needle  so  as  to  prevent  its  vibrations  while  the  instrument 
is  being  carried  from  place  to  place.  A  plumb  is  suspended 
from  the  axis  of  the  transit,  by  means  of  which  its  centre 
may  be  placed  over  a  point  on  the  ground. 

THE    VERNIER. 

The  vernier,  in  the  transit,  is  a  graduated  index  which 
serves  to  subdivide  the  divisions  of  the  graduated  arc  on 
the  lower  plate.  There  are  many  varieties  of  the  vernier, 
but  familiarity  with  one  renders  easy  the  acquaintance 
with  all,  since  the  same  general  principle  is  pervading. 


The  figure  represents  a  common  form.  Let  a  b  be  part 
of  any  graduated  arc,  and  c  d  the  vernier.  It  will  be 
observed  that  the  degrees  on  the  limb  are  divided  into 
spaces  of  15'  each.  Now  if  the  vernier  be  made  equal  in 
length  to  fourteen  of  those  spaces,  and  be  further  divided 
into  fifteen  equal  parts,  it  is  evident  that  each  of  these 
parts  will  contain  14'. 

Then,  if  0  of  the  vernier  coincides  with  any  division  of 
the  limb,  the  first  line  of  the  vernier  to  the  left  will  be 
just  one  minute  behind  the  first  line  of  the  limb  to  the 
left ;  the  second  vernier  line  two  minutes  behind  the  second 
limb  line,  and  so  on ;  so  that  if  the  vernier  be  moved  to 
the  left  over  the  space  of  15'  on  the  limb,  the  lines  from  0 
to  15  of  the  vernier  would  coincide  successively  with  lines 


TO   ADJUST   THE   TRANSIT.  19 

of  the  limb,  and  thus  any  angle  may  be  read  accurately  to 
minutes. 

The  vernier  in  the  figure  reads  48'  to  the  left.  A 
vernier  graduated  decimally  is  much  more  convenient  on 
railway  locations  than  those  with  the  common  graduation 
to  minutes.  This  is  principally  on  account  of  its  adapted- 
ness  to  running  in  curves  when  the  100  feet  chain  is  used. 
The  work  can  be  done  with  more  ease  and  rapidity.  One 
objection  to  it  is  that  the  tables  in  general  use  are  calcu- 
lated for  degrees  and  minutes. 

TO   ADJUST   THE   TRANSIT. 

Place  the  instrument  firmly  at  a,  level  it,  clamp  all 
fast,  and  with  tangent-screw  set  the  cross -hairs  on  the 
point  6,  at  any  convenient  distance.  Reverse  the  telescope 
on  its  axis,  and  fix  another  point  in  the  opposite  direction, 


as  nearly  as  possible  equidistant  from  a.  Now  loose  the 
lower  clamp  and  revolve  the  entire  upper  part  of  the 
instrument  half  way  round  on  its  axis.  Clamp  fast,  and 
having  brought  the  cross-hairs  again  to  coincide  with  6, 
reverse  the  telescope.  If  the  sight  strikes  as  before,  the 
instrument  is  in  adjustment.  If  not,  place  another  point, 
d,  where  it  does  strike,  and  suppose  c  to  be  the  point  pre- 
viously fixed :  the  point  e,  midway  between  d  and  c,  is  then 
in  the  straight  line.  With  the  adjusting  pin  carefully 
place  the  vertical  cross-hair  upon  /,  distant  from  d  one- 
quarter  of  the  space  d  c — with  tangent-screw  set  it  on  e, 
and  reverse  the  telescope.  If  the  points  have  been  cor- 
rectly placed,  and  the  hair  properly  moved,  the  sight  will 
strike  5,  and  the  adjustment  is  complete. 


20 


RAILWAY    CURVES    AND    LOCATION. 


After  finishing  this  adjustment,  the  telescope  may  still 
not  revolve  truly  in  the  meridian.  This  inaccuracy  there 
is  no  method  of  removing  in  the  field.  It  should  be  sent 
to  an  instrument-maker  for  repairs. 


ARTICLE  II. 

\ 

PRELIMINARY   PROPOSITIONS. 


1.  In  any  circle  the  angle  o  cf  at  the  centre,  subtended 
by  the  chord  o  /,  is  double  the  angle  o  af,  at  any  part  of 
the  circumference  on  the  same  side  of  the  chord. 


2.   The  angle  fbe,  formed  by  any  chord  fb,  with  a 
tangent  at  either  extremity,  is  called  a  tangential  angle. 


AVOIDING   OBSTACLES.  21 

and  is  equal  to  half  the  angle  /  c  b  at  the  centre,  or  is 
equal  to  the  angle  /  a  b  at  the  circumference. 

3.  The  exterior  angle  dbf,  formed  at  the  circumference 
bj  the  two  equal  chords  a  b,  b  /,  is  called  a  deflexion  angle, 
and  is  equal  to  the  central  angle  b  c  f,  or  double  the  tan- 
gential angle  e  bf.     df  is  called  the  deflexion  distance, 
and  e  f  the  tangential  distance. 

4.  The  exterior  angle  p  o  m  of  two  unequal  chords,  is 
equal  to  the  sum  of  their  tangential  angles,  or  half  the  sum 
of  their  central  angles. 

5.  The  exterior  angle  i  k  o,  formed  by  tangents,  is  equal 
to  the  central  angle  b  c  o,  subtended  by  the  chord  which 
connects  their  points  of  contact  with  the  curve. 


ARTICLE  III. 

TO   AVOID   AN   OBSTACLE   IN   THE   LINE   OF   TANGENT. 

A  GLANCE  at  the  figure  will  show,  that  having  deflected 
to  c,  and  placed  the  instrument  at  that  point,  the  angle 
hcd  must  be  made  equal  to  twice  dbc,  and  the  distance 


c  d  equal  to  the  distance  b  c.  Still  another  deflection  at 
c?,  equal  to  the  original  angle  turned,  is  necessary  in  order 
to  sight  again  along  the  tangent. 

Should  the   obstruction   be  continuous  a  parallel  line 
may  be  run,  as  from  c  to  /,  by  deflecting  at  c  an  angle 


22  RAILWAY    CURVES   AND   LOCATION. 

equal  to  that  at  5,  and  at  /,  repeating  the  deflection  in 
order  to  strike  tangent. 

If  the  angle  die  exceeds  4°,  and  the  distance  b  c  is 
greater  than  200  feet, — or  even  with  an  angle  of  2|°, 
should  the  distance  be  greater  than  300  feet, — b  c  will 
'differ  sensibly  in  length  from  b  &,  and  a  calculation  of  the 
latter  becomes  necessary.  To  effect  this,  multiply  the 
natural  cosine  of  the  angle  kbc  by  be.  This  result 
doubled  will  give  bJcd,  the  length  proper  along  tangent. 

Thus : — Suppose  kbc,  the  angle  deflected,  to  have  been 
5°,  and  the  distance  b  c  340  feet.  Then  -9962,  the  natural 
cosine  of  5°,  multiplied  by  340,  gives  338-7  for  the  dis- 
tance bJc.  Double  this  makes  bkd  =  677*4,  and  shows 
a  difference  of  2'6  feet  between  b d  and  bed. 


ARTICLE  IV. 

SHOULD   THE   OBSTRUCTION   LIE    ON   THE   OPPOSITE   BANK 
OF   A   STREAM, 

AND  it  is  desirable  on  any  account  not  to  run  the  line 
from  d,  corresponding  to  a  d,  set  the 
instrument  at  a,  in  tangent,  and  deflect 
clear  of  the  obstacle  to  d.  Point  at  d, 
deflect  to  e,  and  point  also  there — 
marking  the  angles  cad,  d  a  e.  Chain 
the  base  de,  and  placing  the  transit  at 
e,  measure  the  angle  dea.  Data  are 
thus  obtained  sufficient  for  the  calcula- 
tion of  the  line  da.  The  object  now  is 
to  find  the  point  c  and  the  angle  d  c  a. 
The  angle  a  d  e  subtracted  from  180°  will  supply  the 


AVOIDING    OBSTACLES. 


-"•r, 

23 


angle  c  da,  so  that  in  the  smaller  triangle  we  have 
obtained  two  angles  and  their  included  side.  The  dis- 
tance cd,  and  angle  dca  readily  follow.  The  transit 
standing  at  e,  c  is  placed,  of  course,  in  the  prolongation 
of  the  base  d  e,  and  the  distance  c  d  is  carefully  set  off 
with  the  rod.  Moving  the  instrument  to  c,  and  turning 
the  angle  ecf  =  180°  —  dca,  we  are  again  in  tangent. 

Example. — Let  cad  =  6°,  dae  =  35°,  d e a  —  42°, 
and  the  base  d  e  —  200  feet.  Then  in  the  triangle  d  e  a 
we  have 

Nat.  sine  d  a  e  —  (-5736)  :  nat.  sine  d  e  a  =  (-6691)  : : 
d  e  =  (200)  :  d  a, 

•6691  x  200 
Wherefore,     d  a  =  — — =  233-3  feet. 


Again,  the  angle  c  d  a  —  77°.  The  angle  d  a  c  being 
=  6°,  ac  d  is  consequently  =  97°,  and  in  the  small  triangle 
we  have,  Nat.  sin.  a  c  d  =  (-9925)  :  nat.  sin.  c  a  d  = 
(-1045)  : :  d  a  =  (233-3)  :  c  d. 

•1045  X  933-3 
Therefore  cd  =  -     — ™^r    -  =  24-564  feet,  and  d  cf 


•9925 


180°  — 97°  =  83C 


NOTE. — A  common  and  convenient  plan  for  triangulat- 
ing a  creek  is  as  per  figure.  Set  the  in- 
strument at  b,  fix  a  point  d  on  the  oppo- 
site shore,  and  making  d  b  c  a  right  angle, 
place  c  at  any  convenient  distance.  Now 
move  to  c,  sight  to  d,  and  making  d  c  a  a 
right  angle  also,  fix  a,  in  the  same  line 
with  b  and  d.  a,  c,  and  d  are  points  in 
the  circumference  of  a  circle  whose  dia- 
meter is  a  d,  a  b  :  b  c  : :  b  c  :  b  d,  and  therefore  b  d  = 


ab. 


24 


RAILWAY   CURVES   AND   LOCATION. 


ARTICLE  V. 

HAVING  GIVEN  THE  ANGLE  edb,  FORMED  BY  THE  INTERSEC- 
TION OF  TWO  STRAIGHT  LINES,  IT  IS  REQUIRED  TO  FIND 
THE  POINT  a  OR  6,  AT  WHICH  TO  COMMENCE  A  CURVE  OF 
GIVEN  RADIUS. 

Draw  the  bisecting  line  d  o.     Then  the  angle  d  o  a  = 

half  the  angle  acb  or  its 
equal  e  d  b ;  and  in  the  tri- 
angle d  c  a,  the  angle  d  a  c 
being  a  right  angle,  we  have 
Rad.  of  1  :  Nat.  Tang. 
dca  :  Rad.  ca:  ad.  There- 
fore a  d  =  Nat.  Tang,  dca 
X  Rad.  e  a. 

Example  1. — Let  e  d  b 
=  48°  and  a  c  =  1460 
feet.  Here  half  the  angle 
edb  or  acb  =  24°,  the 
Nat.  Tang,  of  which  is  P4452 ;  and  multiplying  by  Rad. 
1460,  we  have  650  feet  for  the  length  of  a  d  or  d  b,  the 
tangents. 

2. — If  ad  be  given  and  radius  required, — 
ad  650 


Rad.  = 


=  1460. 


Nat.  Tang,  a  c  d       -4452 

The  following  rules  are  approximate,  and  sufficiently 
correct  for  all  purposes  of  location. 

To  find  the  degree  of  curvature  of  a  b  divide  5730  by 
the  radius  in  feet ;  and  to  find  the  length  of  the  curve  in 
feet  divide  the  angle  acb  (after  reducing  minutes  to  hun- 
dredths)  by  the  degree  of  curvature — the  chord  in  each 
case  being  100  feet  in  length. 


_  TO   TRACE   A   CURVE.  25 

ARTICLE  VI. 

TO   TRACE   A   CURVE  WITH   TRANSIT  AND   CHAIN. 

THE  degree  of  curvature  and  the  angle  to  be  turned  are 
known.  If  the  latter  is  expressed  in  degrees  and  minutes, 
reduce  the  minutes  to  hundredths,  since  the  100  feet  chain 
is  used,  and  divide  the  whole  angle  by  the  degree  of  cur- 
vature. The  quotient  will  be  the  length  of  the  curve  in 
feet,  and  the  P.  T.  is  at  once  ascertained. 


Let  m  a  be  the  tangent,  and  a  the  P.  C.  Place  the 
transit  at  a,  index  reading  0,  and  direct  the  sight  along 
the  tangent  m  a  e.  The  first  deflection  will  be  half  the 
central  angle  subtended  by  the  chord  used,  and  all  the 
stakes  put  in  from  a  will  be  fixed  by  similar  tangential 
deflections.  (Prelim.  Prop.  1.) 

3       - 


26  RAILWAY   CURVES   AND   LOCATION.. 

When  the  point  d  is  reached,  the  angle  dab,  shown 
on  the  index,  will  be  half  the  angle  dbe,  or  its  equal 
a  c  d,  at  the  centre.  Move  the  instrument  to  d,  sight 
back  to  a,  and  turn  to  double  the  index  angle.  The 
telescope  is  now  directed  along  the  tangent  b  d  g,  and 
the  angle  dbe  =  acd  =  dab-\-adb,  reads  on  the 
index.  Note  this  angle  in  the  column  of  tangents  oppo- 
site station  d.  Continue  the  curve  from  this  new  posi- 
tion, precisely  as  was  done  at  a,  and  set  the  point  h. 
Move  to  A,  see  that  the  vernier  has  not  been  disturbed, 
and  sight  back  to  d.  The  index  now  shows  the  angle 
(db  e  -\-  k  d  g\  and  the  object  is  to  turn  the  angle  d  hf, 
i.  e.  repeat  the  angle  fd  h,  as  was  done  before  at  d,  and, 
at  the  same  time,  have  the  whole  angle  (d  b  e  -\-  hfg) 
indicated  on  the  plate.  To  effect  this,  merely  add  this 
angle  fhd  to  the  present  reading.  It  will  be  found  sim- 
pler, in  practice,  to  double  the  entire  angle  thus  far  turned, 
and  subtract  from  the  product  the  last  tangent,  viz.  db  e. 
The  vernier,  turned  to  this  resultant  angle,  will  put  the 
telescope  in  tangent  line  to  h.  And  so  on. 

Example. — "  At  sta.  24  -f-  50  commence  a  4°  curve  to 
the  left  for  35°  12'."  Suppose  this  a  required  duty.  First, 
reducing  minutes  to  hundredths,  we  have  35°-20,  which, 
divided  by  4°,  gives  880  feet  for  the  length  of  the  curve. 
Adding  880  to  24  +  50  it  is  at  once  seen  that  sta.  33  -f  30 
is  the  P.  T. 

Let  a  be  the  P.  C.,  =  24  +  50.  Now  the  deflexion 
angle  being  4°,  the  tangential  angle  is  2°,  with  a  chord  of 
100  feet.  With  a  chord  of  50  feet,  therefore,  the  tan- 
gential angle  is  1°,  and  this  deflexion  from  tangent  m  a  e 
fixes  station  25.  A  deflexion  from  this  latter  point  of  2°, 
the  chord  being  100  feet,  fixes  station  26.  And  so  on. 

When  you  have  fixed  the  point  d,  =  sta.  28,  the  index 
reads  7°.  Move  up  to  station  28,  sight  back  to  the  P.  C., 
and  turn  the  index  to  14°.  This  throws  you  on  tangent 


TO   TRACE   A   CURVE.  27 

Proceed  as  before,  with  the  2°  deflexions,  to  sta.  31,  =  A. 
Move  up,  and  sight  back  to  sta.  28.  The  index  now  reads 
20°.  Multiplying  by  2,  and  subtracting  the  last  tangent, 
we  have  the  reading  of  the  tangent  at  h  =  26° :  we  have 
turned  26°  of  the  curve.  Continue  as  before.  After 
putting  in  sta.  33,  to  find  the  deflexion  which  shall  fix  the 
P.  T.,  33  +  30,  say,  as  100  feet  :  30  feet  : :  2°  :  the  re- 
quired deflexion,  =  36'.  We  may  here  remark  the  great 
convenience  of  an  instrument  graduated  to  hundredths  of* 
a  degree  instead  of  sixtieths.  In  the  present  example  it 
would  be  seen  immediately  that  the  tangential  angle  for 
100  feet  being  2°,  for  1  foot  it  would  be  2  hundredths  of 
a  degree,  and  for  30  feet  it  would  be  60  hundredths. 

Well !  when  the  P.  T.,  =  33  +  30,  is  fixed,  the  index 
reads  30°  36'.  Move  up,  see  that  the  vernier  has  not  been 
disturbed,  and  sight  back  to  sta.  31.  Now  twice  the  index 
reading,  minus  the  last  tangent,  =  61°  12'  —  26°,  = 
35°  12^  the  present  tangent,  which  is  the  final  tangent, 
which  finishes  the  curve. 

The  advantage  of  this  manner  of  running  a  curve  is  that 
the  instrument  shows  at  a  glance  the  work  done,  and  there- 
fore errors  may  be  detected  with  greater  facility.  By 
comparing  at  the  P.  T.  the  total  index  angle  with  the 
distance  run,  the  work  is  tested  at  once. 


}  RAILWAY   CURVES   AND   LOCATION. 

The  above  is  recorded  in  the  field  book  as  follows 


CO  CO  CO  CO 


o 


In  running  compound  and  reversed  curves  the  operation 
is  quite  as  simple  as  the  foregoing.  A  point  is  fixed  at 
the  P.  C.  C.,  or  P.  R.  C.,  and  turning  into  tangent  at 
that  point,  the  second  curve  is  traced  from  this  tangent, 
without  regard  to  what  precedes.  In  reverse  curving,  it 
is  a  good  plan  to  adjust  the  index  in  such  manner  at  the 
P.  R.  C.,  that  when  we  turn  into  tangent  it  will  read  0. 


TO   TRIANGULATE    ON    A    CURVE. 


29 


This  saves  troublesome  work,  and  it  is  advisable  more/.ver 
to  show  in  the  field-book  the  contained  angle  of  each  curve, 
as  well  as  the  test  of  the  two  tangents  with  the  magnetic 
course. 


ARTICLE  VII. 


TO   TRIANGULATE   ON   A   CURVE. 


SET  the  transit  at  a,  and,  as  usual,  sight  back,  and  turn 
into  tangent.  Estimate  the  distance  to  the  farther  bank 
— do  it  liberally — and  make  a  deflexion  around  the  curve, 
corresponding  to  your  estimated  distance.  Fix  a  point  b 
in  this  line.  Measure  any  convenient  angle,  b  a  c,  and  set 


the  point  c.  Move  to  J,  measure  the  base  b  <?,  the  angle 
a  b  c,  and,  before  lifting  the  instrument  calculate  the  line 
b  a.  If  the  angle  turned  from  tangent  to  d  exceeds  4°, 
and  the  distance  is  greater  than  200  feet,  the  chord  a  d 
must  also  be  calculated,  as  per  example,  and  the  difference 
between  this  and  b  a  will  be  the  distance  b  d  to  the  point 
d,  in  the  curve,  which  can  be  fixed  from  b. 

3* 


30  RAILWAY  CURVES   AND   LOCATION. 

Should  J  fall  between  d  and  a  the  operation  is  analogous. 

Example. — Let  a  be  a  point  in  a  6°  curve.  Having  set 
the  transit,  and  turned  into  tangent,  the  distance  to  the 
farther  verge  is  estimated  400  feet.  The  tangential  angle 
for  100  feet  is  3°,  and  to  fix  d,  400  feet  distant,  is  conse- 
quently 12°.  Deflect  this  angle,  fix  a  point  in  line,  and 
complete  the  triangulation,  as  previously  illustrated  in 
Art.  IV.,  p.  22.  Suppose  a  b  found  equal  to  472  feet. 
Now  the  tangential  angle  to  d  =  half  the  central  angle, 
=  feat  =  12°;  and  to  find  the  length  of  the  chord  a  d, 
we  have,  in  the  triangle  efa, 

Kad.  :  Sin.  a  ef : :  e  a  :  a/,  that  is 

Had.  of  1  :  Nat.  Sin.  12°  : :  955-4,  the  Rad.  of  the  6° 
curve  :  half  the  chord  required.  Wherefore  a  d  =  twice 
the  Nat.  Sin.  12°  X  955-4  =  -2079  X  2  X  955-4  == 
397-25  feet.  Subtracting  this  from  472,  we  have  the 
distance,  74-75  feet,  back  to  the  point  in  the  curve.  Move 
the  instrument  to  d,  set  the  index  at  12°,  sight  back  to  a, 
and  turning  to  24°,  the  telescope  is  in  tangent.  A  deflexion 
of  3°  will  fix  the  next  station. 

NOTE. — In  this  case,  if  preferred,  a  third  proportional 
might  be  formed  with  the  chord  of  crossing,  as  shown  in 
the  note  to  Art.  IV. 


TO   CHANGE    THE   ORIGIN    OF   A    CURVE. 


31 


ARTICLE  VIII. 


TO  CHANGE  THE  ORIGIN  OF  ANY  CURVE,  SO  THAT  IT  SHALL 
TERMINATE  IN  A  TANGENT  PARALLEL  TO  A  GIVEN  TAN- 
GENT. 

LET  eZ/be  the  located  curve,  terminating  in  a  tangent/^, 
and  the  nature  of  the  ground  requires  that  it  should  ter- 
minate in  the  tangent  e  i,  parallel  to  /  k.  At  /,  the  tele- 
scope being  directed  along  the  tangent  fk,  turn  to  the 


right  an  angle  equal  to  the  central  angle  d  bf,  previously 
turned  to  the  left  on  the  curve.  This  will  direct  the  tele- 
scope along  ef,  parallel  to  d  I  Measure  ef,  and  go  back 
on  the  tangent,  d  I,  a  distance,  c  d,  equal  to  it.  The  curve, 
retraced  from  c  and  consuming  the  same  angle,  -will  termi- 
nate tangentially  in  e  i.  An  example  in  this  case  is  not 
necessary. 


RAILWAY   CURVES   AND   LOCATION. 


ARTICLE  IX. 


TO  CHANGE  A  P.  C.  C.  SO  THAT  THE  SECOND  CURVE  SHALL 
TERMINATE  IN  A  TANGENT  PARALLEL  TO  A  GIVEN  TAN- 
GENT. 

LET  a  b  d  be  the  compound  curve,  located  and  terminat- 
ing in  the  tangent  d  h.  Continue  the  larger  curve  to  e, 
and  from  e,  with  radius  e  I  =  kb,  describe  the  curve  ef, 
terminating  tangentially  in  f  g,  parallel  to  dh.  From  c, 
the  centre  of  the  larger  curve,  let  fall  upon  fg  the  perpen- 


dicular eg,  and  fill  up  the  figure  as  above.  Call  the  radii 
respectively  R  and  r,  the  angle  b  k  d,  or  its  equal,  k  c  m, 
x,  and  the  angle  e  If,  or  its  equal,  lcn,y.  Let  the  dis- 
tance if,  or  h  <7,  be  named  D.  Now  the  line  c  g  is  made 
up  of  the  lines  cm  -{-  mh  -\-  hg,  i.  e.,  eg  =  cosin.  x 
(  R  —  r)  -f-  r  -f-  D.  c  g  is  also  made  up  of  the  lines  c  n 
+  ng,  i.  e.,  eg  =  cosin.  y  (R  —  r)  -f-  r.  Therefore  cosin. 
z(R  —  r)-fr  +  D  =  cosin.  y  (R  —  r)  +  r,  and  reducing, 


TO   CHANGE   A   P.  C.  C.  33 

cosin.  x  (R  —  r)  -4-  D 
cosm.  y  =  --  ^5  -  -  -  ;  so  that  the  distance  if. 

-  (-CV  -  T) 

or  Jig,  measured  rectangularly  between  the  two  tangents, 
being  added  to  the  nat.  cosin.  x,  will  give  the  nat.  cosin.  of 
the  angle  e  If,  to  be  turned  on  the  smaller  curve.  The 
angle  y,  subtracted  from  the  angle  x,  gives  of  course  the 
angle  b  c  e,  to  be  advanced  on  the  larger  curve  ;  or,  divid- 
ing this  angle  by  the  degree  of  curvature  of  a  b,  we  find 
the  distance  from  b  to  e  the  P.  C.  C.  proper. 

If  ef  be  the  second  curve  located,  and  the  tangent  to  be 
touched  lies  within,  it  is  evident  that  we  must  retreat  upon 
the  large  curve,  and,  by  subtracting  D  from  the  cosine  of 
the  angle  y,  we  obtain  the  cosine  of  the  angle  x. 

Example.  —  Suppose  a  b  a  3°  curve  located,  and  com- 
pounding, at  b,  into  a  6°  curve,  which  latter  is  continued 
to  the  right  through  an  angle  of  42°.  At  the  P.  T.  we 
discover  that  the  proper  tangent  is  64  feet  to  the  left. 
We  must  throw  our  curve  out,  then  —  we  must  advance  on 
the  3°  curve  a  certain  distance.  How  to  find  this  distance  : 
The  radius  of  a  3°  curve  =  1910  ;  the  radius  of  a  6°  curve 
=  955-4;  R  —  r,  therefore,  =  954-6.  The  nat.  cosin. 
42°  =  '7431.  Now,  by  the  formula  just  obtained,  we 
cos,  x  (R  —  r}  +  D  (-7431  X  954-6)  +  64  _ 
~-~  ~~ 


•8101  =  nat.  cosin.  35°  53'.  Subtracting  this  from  42°, 
we  have  6°  07',  the  angle  to  be  advanced  on  the  3°  curve  ; 
or,  reducing  minutes  to  hundredths,  and  dividing  by  3°,  we 
find  204  feet,  the  distance  from  b  to  the  correct  P.  C.  C: 


34 


RAILWAY   CURVES   AND   LOCATION. 


ARTICLE  X. 

SHOULD   THE   SECOND   CURVE   BE   ONE   OF   LONGER   RADIUS 
THAN   THE   FIRST, 

OUR  illustration  takes  simpler  form,  and  the  application 
of  D  varies  vic§  versa. 


See  figure,  analogous  to  that  of  the  previous  problem. 
Here  of,  eg  are  equal  and  parallel  radii;  fh  a  perpen- 
dicular connecting  them.  Draw  its  fellow,  en.  Then, 
nf  =  c  h,  and,  consequently,  h  g  =  6  n.  Again,  letting 
k  m  fall,  perpendicular  to  c  g,  we  have  c  m  =  n  I,  and  o  I 
=  c  m  -f-  o  n ;  i.  e.,  cosin.  y  (R  —  r)  =  cosin.  x  (R  —  r) 
+  D. 

We  observe  that,  with  a  curve  of  this  nature,  in  order 
to  throw  the  line  farther  out,  it  is  necessary  to  go  back, 
toward  b ;  or,  having  located  to  g,  if  the  object  tangent  lie 
within,  we  must  advance  toward  e. 

Example. — Suppose  a  b  a  5°  curve,  b  the  P.  C.  C,,  and 
b  g  a  2°  curve.  Setting  the  instrument  at  g,  the  P.  T., 
and  turning  into  tangent,  we  find  that  we  are  a  distance 
hg,  =  53  feet,  too  far  to  the  left.  The  first  question  is, 
•what  angle  have  we  turned  on  the  second  curve.  Let  it 


TO   CHANGE   A   P.  C.  C.  35 

be  28°.  Now  we  know,  that,  in  order  to  strike  farther  to 
the  right,  we  must  advance  on  the  5°  curve.  Consequently, 
I)  must  be  added  to  the  cosine  of  28°,  to  give  us  the  cosine 
of  the  proper  angle  for  the  2°  curve  ;  and  the  difference 
between  28°  and  this  newly  found  angle  will  be  the  angle 
we  are  to  advance  on  the  5°  curve.  Thus  :  —  the  rad.  of  a 
5°  curve  =  1146  feet,  that  of  a  2°  curve  =  2865  feet, 
and  their  difference  =  1719  feet.  The  nat.  cos.  of  28° 
=  -8829.  Then  ('8829  X  1719)  +jg  =  .9m|  =  nat 

cos.  23°  58*.  This,  subtracted  from  28°,  leaves  4°  02',  = 
80  feet,  from  b  to  the  correct  P.  C.  C. 

Synopsis  of  the  preceding  formula. 

Call  D  the  distance  between  tangents  as  before,  a  the 
angle  of  the  second  curve  located,  and  b  the  angle  of  the 
same  curve  to  be  substituted  for  it. 

FIRST,  when  the  second  curve  has  the  smaller  radius  — 
Tangent   falling  within  the  point,  cosine  b  =• 
cos.  a  (R  —  r)  -f-  D 

(R-r)~ 

Tangent  falling  without  the  point,  cosine  b  = 
cos.  a  (R  —  r)  —  D 

(*-') 

SECOND,  when  the  second  curve  has  the  larger  radius  — 
Tangent  falling  within  the  point,  cosine  b  = 
cos.  a  (R  —  r)  —  D 

(R-r)  ' 

Tangent  falling  without  the  point,  cosine  b  = 
cos.  a  (R  —  r)  +  D 


Very  little  attention  will  familiarize  these  formulae,  and 
render  the  field  practice  easy. 


RAILWAY   CURVES   AND   LOCATION. 


ARTICLE  XL 

HAVING  LOCATED  THE  COMPOUND  CURVE  a  I  d,  TERMINAT- 
ING IN  THE  TANGENT  df,  IT  IS  REQUIRED  TO  FIND  THE 
P.  C.  C.  6,  AT  WHICH  TO  COMMENCE  ANOTHER  CURVE  OF 
GIVEN  RADIUS,  WHICH  SHALL  ALSO  TERMINATE  TAN- 
GENTIALLY  IN  df. 

PLOT  the  curves  as  per  figure.  From  c  let  fall  c  g,  per- 
pendicular to  the  tangent,  df.  From  k  and  z,  the  lesser 
centres,  drop  k  m,  i  ?,  perpendicular  to  c  g.  Call  the  great 
radius  B,  the  smaller  radius  r,  and  the  intended  radius  of 


the  second  curve  /.  Likewise  name  h  k  d,  the  angle  of 
the  small  curve  located,  #,  and  lie  the  angle  to  be  found 
for  the  proposed  curve,  y.  Now,  the  tangent  df,  and  the 
curve  a  b,  lying  unstirred,  the  line  c  g  is  an  unvarying  dis- 
tance, and  it  is  made  up  of  the  lines  c  m  -f  mg,  i.  e.,  eg 
=  (B  —  r)  cosin.  x  +  r.  It  also  consists  of  the  lines  c  I 


COMPOUND   CURVES  37 

-f-  Iff,  i.  e.,  off  =  (R  —  r'}  cosin.  y  +  /,  and,  reducing, 

(R  —  r)  cosin.  x  -4-  r  —  / 

nat.  cosin.  y  = '—^ -^ .     This,  there- 

(K,  —  r ) 

fore,  is  the  formula  by  means  of  which  we  can  ascertain 
the  point  5,  as  follows : — 

Example. — Imagine  a  2°  curve,  a  h,  compounding  into 
a  6°  curve,  h  d,  which  terminates  at  d,  in  the  tangent  df. 
^The  tangent  lies  well ;  the  curve  a  h  likewise ;  but  it  is 
desired  to  throw  the  line  to  the  left,  on  better  ground, 
between  d  and  h,  by  means  of  an  intercalary  4°  curve. 
We  wish,  then,  to  know  the  distance,  h  6,  back  to  the  new 
P.  C.  C. 

The  radius  of  a  2°  curve  =  2865  feet,  of  a  6°  curve  = 
955-4  feet,  and  their  difference  (R  —  r)  =  1909-6.  The 
radius  of  a  4°  curve  =  1433,  and  the  difference  (R  —  r') 
is,  therefore,  1432  feet.  Let  h  k  d,  the  angle  turned  on 
the  6°  curve,  be  41°,  the  nat.  cos.  of  which  =  -7547. 

T,        (-7547  X  1909-6)  +  955-4  —  1433        RYOft 

Inen,  -  ^32  *bTJ°>  = 

nat.  cosin.  47°  43'.  Subtracting  41°,  we  have  6°  43',  the 
angle  h  c  6.  Reducing  minutes  to  hundredths,  and  divid- 
ing by  2°,  we  find  336  feet  to  be  the  distance  from  h  to  b. 
A  4°  curve  of  47°  43',  traced  from  this  latter  point,  will 
terminate  in  the  tangent  ef. 


38 


RAILWAY   CURVES  AND   LOCATION. 


ARTICLE  XII. 


IF  THE  LATTER  CURVES  HAVE  LARGER  RADII  THAN  THE 
FIRST, 

THE  solution  retains  its  shape  and  simplicity. 

Draw  the  figure  as  above,  and,  for  the  sake  of  uniformity, 
name  the  radii  as  before.  The  curve  a  5,  and  tangent  df, 
being  constant,  the  distance  m  g,  or  d  n,  is  here  constant 
Call  it  A.  Now  A,  in  the  first  place,  is  equal  to  d  i  —  n  i 
i.  e.,  =;  r  —  (r  -r—  R)  cosin.  x\  and,  in  the  second  place 


it  is  equal  to  g  c  —  m  c,  =  i1  —  (/  —  R)  cosin.  y ;  where- 
fore r  —  (r  —  R)  cosin.  x  =  /  —  (/  —  R)  cosin.  y,  and 


consequently  cosin.  y 


_(r  —  R)  cosin.  x  +  /  —  r 


(/-R) 

Example. — Suppose  a  b  a  7C  curve,  compounding,  at  b, 
into  a  5°  curve,  5  d,  which  latter  subtends  an  angle  of  38°, 
and  terminates  in  the  tangent  df.  We  wish  to  substitute 
a  terminal  2°  curve,  hg,  and  to  know  the  position,  h,  of 
the  new  P.  C,  C. 

The  radius  of  a  7°  curve  =  819  feet  ==  R.     r  and  r', 


TO   CHANGE   A   P.  R.  C. 


39 


the  radii  respectively  of  5°  and  2°  curves,  are  equal  to 
1146,  and  2865  feet,  r  —  R,  therefore,  =  327,  and 
/  —  R  =  2046  feet.  The  nat.  cosin.  of  38°  =  -788. 

mr,  k    '   A    '  *  1        C788  X  327)  +  2865  —  1146 

Then,  by  the  formula,  * —  2fi46 

•9661,  =  the  nat.  cosin.  of  14°  57',  the  angle  to  be  turned 
on  the  2°  curve.  Subtracting  this  from  38°,  we  have  23° 
03',  the  angle  to  be  continued  on  the  7°  curve.  Reducing 
minutes  to  hundredths,  and  dividing  by  the  degree  of  curva- 
ture, 7°,  we  find  329  feet,  the  distance  from  b  to  the  new 
P.  C.  C.  h. 


ARTICLE  XIII. 


TO  CHANGE  A  P.  11.  C.,  SO  THAT  THE  SECOND  CURVE  SHALL 
TERMINATE  IN  A  TANGENT  PARALLEL  TO  A  GIVEN  TAN- 
GENT. 

LET  a  d  I  be  the  reverse  curve,  located  and  terminating 


1 K 


in  the  tangent  I L     Call  the  radius  c  5,  R,  and  the  radius 
6  e,  r.     Suppose  ig  the  given  tangent.     At  a  distance 


40  RAILWAY   CURVES  AND   LOCATION. 

from  it  equal  to  *  e,  the  radius  of  the  second  curve,  draw 
the  parallel  line,  op.  With  c  as  a  centre,  and  radius  cf, 
==  R  +  r,  describe  the  integral  curve,  fe,  cutting  op  in  e. 
e,  then,  is  the  centre  of  the  curve  adjusted. 


Application. 

Place  the  transit  at  the  P.  T.,  ?,  and  turn  into  a  tan- 
gent, Im,  parallel  to  dn,  the  common  tangent  of  the  two 
curves  at  d.  Unless  some  wide  mistake  has  been  made, 
the  distance  Ik,  measured  along  this  line  to  ig,  the  tan- 
gent proper,  will  be  about  equal  to  the  distance  ef,  and 
we  shall  have  the  proportion,  cfife  : :  c  d  :  db,  i.  e.,  R 

-f-  r  :  ef  : :  R  :  d  5,  which  gives  -^—. ,   as  a  simple 

XV   — J~   7* 

formula  for  finding  the  distance  back  from  d  to  5,  the  cor- 
rect P.  R.  C.  This  rule,  though  sufficiently  true  for  most 
cases,  is  not  mathematically  justifiable.  It  will  be  seen 
that  ef,  or  its  equal  i  I,  the  distance  we  wish  to  measure, 
is  a  curving  distance,  part  of  the  circumference  of  a  circle 
concentric  with  a  b.  Its  radius  is  (R  +  r),  therefore  its 

5730 
degree  of  curvature  =  ,„         .,  or,  more  simply,  equals 

the  product  of  the  degrees  of  curvature  of  the  curves  com- 
posing the  reverse,  divided  by  their  sum.  To  be  strictly 
accurate,  then,  set  the  instrument  at  I,  turn  into  tangent 
I  m  as  before,  and  trace  the  curve  i  I,  until  it  strikes  the 
tangent  ig.  The  angle  which  il  subtends,  being  divided  by 
the  degree  of  curvature  of  a  b,  will  give  the  distance,  d  b, 
to  the  P.  R.  C.  proper.  The  curve  retraced  from  b,  will 
terminate  tangentially  in  ig,  and  its  angle,  b  e  i,  will  be 
equal  to  dfl  —  deb. 

Example. — Let  a  d  I  be  a  reverse  curve,  composed  of  a 
3°  curve,  a  d,  and  a  6°  curve,  dL  Let  the  angle  dfl  be 
equal  to  52°,  and  suppose  the  distance  If  to  have  been 


WHEN    P.  C.    IS   INACCESSIBLE. 


41 


found  34  feet.  Being  part  of  a  2°  curve,  it  therefore  sub- 
tends a  central  angle  of  41'.  This  corresponds  to  a  dis- 
tance of  23  feet,  to  be  gone  back  on  the  3°  curve,  and 
52°-00  —  41'  =  51°  19',  the  angle  to  be  turned  from  h 
on  the  6°  curve,  in  order  to  strike  the  tangent  ig. 


ARTICLE  XIV. 

HOW   TO    PROCEED   WHEN   THE    P.  C.    IS  INACCESSIBLE. 

IN  the  figure,  drawn  to  illustrate  this  case,  let  c  be  the 
point  of  curvature,  c  a  the  tangent,  and  c  k  e  the  curve. 
Now  the  angle  dee,  included  between  the  tangent  and 


any  chord,  as  c  e,  fixing  the  point  e,  is  known.  Make  c  b 
along  tangent,  equal  to  c  e,  and  connect  be.  If  a  circle 
were  now  described  from  c  .as  a  centre,  with  radius  e  e  or 
c  b,  d,  e,  and  b,  would  be  points  in  its  circumference,  and 


42  RAILWAY    CURVES   AND   LOCATION. 

the  angle  d b e  at  once  proven  equal  to  half  the  angle  dee. 
With  proof  precisely  similar,  d  c  e  =  half  of  d  g  e,  and, 
consequently,  d  b  e  is  equal  to  one-fourth  of  the  central 
angle  subtended  by  the  chord  c  e. 

Example. — Suppose  c  to  be  the  inaccessible  point  of 
curvature  of  a  6°  curve,  eke.  It  is  concluded  to  run  to 
the  third  station,  e.  First  we  must  calculate  the  length 
of  the  chord  c  e.  The  angle  d  c  e  =  9°,  and  from  Art. 
VII.  we  have 

Had.  of  1  :  9554  : :  nat.  sin.  9°  =  -1564  :  e-£,  whereby 

m 

ec  is  shown  equal  to  298-8.  Place  the  transit  then  at  5, 
298-8  feet  distant  from  the  P.  C.,  and  deflect  to  the  left 
an  angle  of  4°  30',  equal  to  half  the  angle  dee.  This  is 
in  line  to  e,  and  b  e  must  likewise  be  calculated  as  follows : 

In  the  triangle  b  c  h  we  have 

Bad.  of  1 :  nat.  cosin.  4°  30'  ==  -9969  : :  b  c  =  298-8  :  b  h 

be 
—  -^-,  whereby  b  e  is  shown  equal  to  595-7  feet.    Arriving 

28 

at  e,  the  index  reads  4°  30'.  Sight  back  to  5,  turn  to  18°, 
and  the  telescope  will  be  in  tangent.  Suppose,  however, 
that  having  reached  /,  100  feet  from  e,  this  latter  point  is 
also  found  inaccessible.  We  find  k  a  different  point  in  the 
curve,  thus:— The  angle/e^  =  18°  —  4°  30'  =  13°  30', 
and  the  tangential  .angle  g  e  k  =  3°.  Consequently  the 
angle  fek  =  10°  30',  and,  drawing  the  bisecting  line  e  z, 
we  have,  in  the  triangle  efi, 

Had.  of  1  :  nat.  sin.  5°  15'  =  -0915  : :  */  =  100  :  /» 
=  9-159  feet.  Therefore  fk  =  18-318  feet,  and  the 
angle  efk  =  90°  —  5°  15'  =  84°  45'.  At  /  deflect  this 
angle  to  the  right,  and  measure  the  distance  fk  carefully 
with  the  rod.  At  k,  sighting  back  to  /,  and  turning  the 
equal  angle  fk  e,  the  telescope  will  be  directed  to  e,  and 
the  curve  may  be  continued. 

If  it  is  inconvenient  to  run  the  line  b  e,  the  point  e  may 


TO   AVOID   OBSTACLES   IN    THE   LINE   OF   CURVE.  43 

be  reached  thus: — Fix  the  P.  C.     Find  the  tangential 
distance  d  e,  corresponding  to  the  angle  dee.     Carefully 
with  the  rod  lay  off  b  Z,  equal  to  it,  at  right  angles  to  b  e 
Set  the  transit  at  I,  and,  in  line  with  c,  put  in  e. 
The  distance  b  I  should  not  exceed  10  or  12  feet. 

NOTE. — The  foregoing  illustrations  will  apply  when  the 
P.  T.  is  likewise  inaccessible. 


ARTICLE  XV. 

TO  AVOID   OBSTACLES   IN   THE   LINK  OF  CURVE. 

LET  b  k  h  be  the  curve.  We  can  either  follow  the  tan- 
gents hd,  db,  or  trace  a  parallel  curve,  g  a,  within  the 
first ;  which  tracing  is  effected  thus  : — Set  the  instrument 


at  A,  the  P.  C.,  and  offset  any  distance  h  g,  at  right  angles 
to  the  tangent  h  d.  It  will  be  observed  that  as  the  dis- 
tance h  g  increases,  the  distance  g  a  decreases,  whilst  the 
angle  subtended  by  g  a  remains  equal  to  that  subtended 


44  RAILWAY   CURVES   AND   LOCATION. 

by  h  b ;  i.  e.,  our  deflexions  on  the  offset  curve  stand 
unchanged,  but  the  corresponding  chords,  g  /,  f  e,  &c., 
are  less  than  their  equivalents,  h  i,  i  k,  &c.,  along  h  b. 
To  find  their  length,  h  z,  i  k,  &c.,  being  equal  to  100 
feet,  we  have  the  proportion,  c  h  :  eg  : :  h  i  :  x ;  i.  e., 
K  :  rad.  —  h  g  : :  100  feet  :  x,  where  x  symbols  the  un- 
known chord.  Now  set  the  transit  at  g,  turn  into  tangent 
parallel  to  h  d,  and  with  the  shortened  chord,  fix/  e  a. 
Rectangularly  to  the  tangents  at  these  points,  and  distant 
h  g,  will  be  «,  k,  b  of  the  curve  proper. 

Example. — Let  5  h  be  a  4°  curve,  and  the  offset  distance 
85  feet.  The  radius  then  is  1433,  and 

1433  :  1348  : :  100  :  94,  the  short  chord. 

To  follow  the  tangents,  suppose  the  angle  b  c  h  =  42°. 
Then  by  Art.  V.  we  find  the  tangent  h  d  =  550  feet, 
which  distance  we  duly  measure,  and,  at  d,  deflecting  42°, 
lay  off  an  equal  distance  to  5,  the  point  of  tangency. 


fIND   THE    RADII    OP   REVERSE   CURVE. 


45 


ARTICLE  XVI. 

HAVING  GIVEN  THE  ANGLES  d  b  k,  m  k  I,  AND  THE  DISTANCE 
b  k,  IT  IS  REQUIRED  TO  FIND  THE  RADII  C  e,  ef  OF  THE 
EASIEST  REVERSE  CURVE  WHICH  SHALL  UNITE  ad,  km. 

THE  angle  d  b  e  is  equal  to  the  angle  ace,  half  of  which 
is  b  o  e.     So  likewise  ef  k  is  equal  to  half  of  I  km. 


Then,  [nat.  tang,  b  c  e  +  nat.  tang,  eflc]  :  nat.  tang. 
o  c  e  : :  b  k  :  b  e,  and  b  k  —  b  e  =  e  k.  Wherefore  rad.  c  e 

be  ek 

=  — — , —  and  rad.  ef  = ,-7-. 

nat.  tang,  bee  nat.  tang,  kj  e 

Example. — Suppose  the  angle  d  b  e  =  54°  30',  the 
angle  Ikg  =  33°  20',  and  the  distance  b  k  =  832  feet. 

Therefore  the  angle  bee  =  27°  15',  the  nat.  tang,  of 
which  is  -5150,  and  the  angle  efk  =  16°  40',  the  nat. 
tang,  of  which  is  -2994.  The  sum  of  the  tangents  =  -8144. 
Then,  to  find  b  e,  we  have 

As  -8144  :  -5150  : :  832  :  526,  and  subtracting  this  from 
b  k,  we  have  e  k  =  306  feet. 

526  306 

Again,  the  radius  c  e  =  TFTH?  and  the  radius  ef  —  -2994' 

=  1022  feet. 


